The Particle-in-Cell (PIC) method has a long history dating back to the 1950s. The method was first developed by computational physicists who were interested in simulating the behavior of plasmas in fusion reactors. Since then, the PIC method has evolved and become a widely used numerical method in a variety of fields, including plasma physics, particle accelerators, semiconductors, and many others. The method has undergone many improvements and advancements. Today, PIC simulations play a crucial role in understanding the behavior of complex systems and have become an important tool for designing and optimizing various devices and processes.

In PIC simulations, particles like electrons and ions are modeled as discrete entities that move in continuous fields that are calculated on a computational mesh. The motion of each particle is calculated based on its interactions with other particles and electromagnetic fields. The properties of each particle, such as its mass, charge, and position, are stored in arrays. The particles are moved in time using numerical integration techniques. One of the key steps in a PIC simulation is the deposition of charges on the grid. The charge deposition is performed by assigning the charge of each particle to the grid points that surround it. This is done using a weighting function that distributes the charge over multiple grid points. The charge density on the grid is then used to calculate the electric and magnetic fields. After the electromagnetic field is calculated, the field strength is interpolated to the position of each particle and the acting Lorentz force is calculated. As a consequence each particle experiences the corresponding electromagnetic acceleration.

#### Choice of the field equations

For the calculation of the field, two approaches are available: electromagnetic and electrostatic. In the electromagnetic approach, the electric and magnetic fields are coupled and calculated by solving Maxwell’s equations. This approach is very accurate and can handle most electromagnetic phenomena, but is computationally very expensive. In the electrostatic approach, the electric field is calculated using Poisson’s equation, where the magnetic field is assumed to be zero. This approach is simpler and computationally faster, but may not be applicable to some systems, where the magnetic fields due to the particle movement cannot be neglected anymore. Examples of simulations using both approaches can be found within our application categories Surface & Vacuum and Electromagnetics.

#### Frequently Asked Questions

##### What are the main computational challenges and limitations of Particle-in-Cell simulations?

One of the main challenges in PIC simulations is managing the computational resources required to simulate large numbers of particles and complex interactions over significant time periods. This includes dealing with the curse of dimensionality in three-dimensional simulations while ensuring numerical stability by resolving the Debye length and plasma frequency. Furthermore, boundary conditions for gas-surface interactions such as secondary electron emission add unknowns and complexity.

##### What numerical integration techniques are commonly used in Particle-in-Cell simulations?

Techniques vary from simple first-order Euler methods, which are straightforward but less accurate, to more advanced second-order methods such as the Leapfrog integration (or Boris-Leapfrog), to complex fourth-order Runge-Kutta methods, which offer higher accuracy at the cost of increased computational effort. The choice of method affects not only the accuracy and stability of the simulation but also its computational cost. Several of the mentioned time discretization techniques are available in PICLas.

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